In this paper, we prove that any perfect complex of $D^{\infty}$-modules may be
reconstructed from its holomorphic solution complex provided that we keep track of the
natural topology of this last complex. This is to be compared with the reconstruction
theorem for regular holonomic $D$-modules which follows from the well-known Riemann-Hilbert
correspondence. To obtain our result, we consider sheaves of holomorphic functions as
sheaves with values in the category of ind-Banach spaces and study some of their
homological properties. In particular, we prove that a K\"{u}nneth formula holds for them
and we compute their Poincar